pentatonicSize
Pentatonic scale size is fixed at five notes inside the phi-derived musical structure. Workers on Fibonacci relations among scale lengths cite this constant to reach the twelve-tone octave total. The definition is a direct natural-number assignment with no further computation.
claimThe pentatonic scale size is defined as five: $5$.
background
The Aesthetics.MusicalScale module derives the twelve-tone equal-temperament scale from the golden ratio through optimal frequency ratios and circle-of-fifths closure. Upstream, the scale function returns phi raised to integer power k while the has class ranks spectral peaks by successive phi-powers. This definition supplies the five-note count among the phi-related sizes (5, 7, 19, 31) listed in the module predictions.
proof idea
The declaration is a one-line definition that directly assigns the constant five.
why it matters in Recognition Science
This supplies the five in the Fibonacci relation five plus seven equals twelve used by the pentatonic_diatonic_fib and scale_fibonacci theorems. It fills the module prediction that scale sizes five and seven carry phi-related structure, supporting the emergence of twelve semitones from phi^5 scaling in the Recognition framework.
scope and limits
- Does not derive the value five from the forcing chain T0-T8.
- Does not prove consonance or optimality properties of the pentatonic scale.
- Does not link the size to explicit rung values on the phi-ladder.
Lean usage
theorem pentatonic_example : pentatonicSize = 5 := rflformal statement (Lean)
168def pentatonicSize : ℕ := 5
proof body
Definition body.
169
170/-- The diatonic scale has 7 notes. -/